International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 5, Issue 2, 2017, PP 11-15 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) DOI: http://dx.doi.org/10.20431/2347-3142.0502002 www.arcjournals.org Dually Normal ADL S. Ravi Kumar 1 Reader, Dept. of Mathematics M.R.College(A), Vizianagaram, India G.C.Rao 2 Department of Mathematics Andhra University Visakhapatnam, India Abstract: The concept of Normal ADL was introduced in my earlier paper Normal Almost Distributive Lattices[2]. In this Paper, we introduce the concept of a dually normal ADL and we characterize Dually Normal ADLs in terms of dual annihilators. Throughout this Chapter we consider ADL R means an ADL with at least one maximal element. Keywords: Almost Distributive Lattice(ADL), Normal ADL, Dual annihilators, Dually Normal ADL. Introduction The concept of Normal lattice was introduced by W.H.Cornish [4] as a distributive lattice with 0 in which every prime ideal contains a unique minimal prime ideal. In [2], we introduced the concept of an ADL R being normal through its principal ideal lattice PI R. That is, an ADL R with 0 is a Normal Almost Distributive Lattice if its principal ideal lattice PI R is a normal lattice. 1. DUAL ANNIHILATORS In this section, we introduce a dual annihilator and study some of its properties. First we start with the following definition. 1.1. Definition : Let R be an ADL with maximal elements and S be any non-empty subset of R. Define ( S ) { x R s x is maximal for all s S }. On routine verification, we can prove the following result. 1.2. Lemma : For any a, b R, if a b is a maximal element in R then for any x R, the elements x a b and a x b are also maximal in R. In the following theorem, we prove that the set ( S ) is a filter in R. 1.3. Theorem : For any non-empty subset S of R, the set ( S ) is a filter of R Proof : Let S be any non-empty subset of an ADL R and a, b ( S ). Then for any s S, the elements s a, s b Consider the element s ( a b), for some s S. Now, for any x R,[ s ( a b)] x = ( s a) ( s b) x = ( s a) x) = x. Therefore a b ( S ). Again, let x R and s ( S ). Clearly, s a is maximal for every s S, and hence the element s ( x a) is maximal for every x R. Therefore, x a ( S ). Therefore ( S ) is a filter in R. In the following, we define a dual annihilator in an ADL R. ARC Page 11
S. Ravi Kumar & G.C.Rao 1.4. Definition : If S {} s in the above theorem, then we write ( S ) ( s) instead of ({ s}) we have ( s) { x R s x is maximal in R }. Clearly, () filter as a dual annihilator of s in R. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 12 s and is a filter in R and we call this In the following theorem, we prove some important properties of dual annihilators in an ADL R 1.5. Theorem : Let R be an ADL with maximal element. Then for any ab, in R Proof : 1). a b ( a ) ( b ) 2). ( a b ) ( b a ) 3). ( a b ) ( b a ) 4). ( a b ) ( a ) ( b ) 5). ( a ) ( b ) ( a b ) 6). a ( x ) ( x ) ( a ) 7). a [ b ) ( b ) ( a ) 8). ( a ] ( b ] ( b ) ( a ) (1) : Let a, b R such that a b. Then a b b b a. Now, x ( a ) a x is maximal in R b a x is maximal in R Therefore ( a ) ( b ) b x is maximal (sinceba b ) x (2) : Let ab, be any two elements of R. Now, x ( a b) a b x is maximal in R b a x is maximal in R x ( b a). Therefore ( a b) ( b a) (3) : Let ab, be any two elements of R. Now, x ( a b) ( a b) x is maximal in R ( b a) x is maximal in R x ( b a). Therefore ( a b) ( b a) (4) : Let ab, be any two elements of R. b We have a b b ( a b) ( b ) (from (1)) Similarly, b a a ( b a) ( a ) ( a b) ( a ) (from (3)). Therefore ( a b) ( a ) ( b ). Again x ( a ) ( b ) x( a ) and x Hence ( a b) ( a ) ( b ) b a x and b x are maximal in R x a and x b are maximal in R ( x a) ( x b) is maximal in R x ( a b) is maximal in R ( a b) x is maximal in R x ( a b) Therefore ( a ) ( b ) ( a b).
Dually Normal ADL (5) : Proof follows from (1) and (2). (6) : Let ax, be any two elements of R. Suppose a International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 13 x Now t ( x ) s t is maximal for all s ( x ). a t is maximal in R ( since a ( x ) ) t ( a ). Therefore ( x ) ( a ) (7) : Let ab, be any two elements of R. Suppose a [) b. Then a b a. Now t ( b ) b t is maximal in R (8) : Proof follows from (7). ab t is maximal a t is maximal in R (since a b a) t ( a ). Therefore ( b ) ( a ). 2. DUALLY NORMAL ADLS In this section we introduce a dually normal and we characterize dually normal ADLs in terms of dual annihilators and in terms of maximal elements. First we define a dually normal ADL in the following. 2.1. Definition : An ADL R with maximal element is said to be dually normal if for any prime filter F of R, C( F) { xr y x is maximal, for some y F is a prime filter of R. In the following theorem, we characterize a dually normal ADL in terms of dual annihilators.\\ 2.2. Theorem : Let R be an ADL with maximal elements. Then the following are equivalent. Proof : 1). R is dually normal. 2). For any x, y R, if x y is maximal then ( x) ( y) R 3). For any x, y R, ( x y) = ( x) ( y) (1) (2) : Assume that R is a dually normal ADL. Let x, y R such that x y is maximal. We have to prove that ( x) ( y) R. Suppose ( x) ( y) R. Then there exists a maximal filter G of R such that ( x) ( y) G. Since G is a prime filter and R is dually normal, we have C(G) is a prime filter of R. Therefore x y C( G) implies that either x C( G) or y C( G). If x C( G), then by definition, there exists some t G such that x t is maximal and hence t ( x) G. This is a contradiction. Therefore x G. Similarly, we get y G. Thus we get x y G. This is a contradiction. Therefore we get ( x) ( y) R. (2) (3) : Let x, y R. We have to prove that ( x) ( y) = ( x ) From (5) of Lemma 1.2, we have ( x) ( y) ( x y). Now, t ( x y) x y t is maximal t x y is maximal t xt y is maximal ( t x) ( t y) = R from condition (2). y
S. Ravi Kumar & G.C.Rao Now t R t s 1 s 2 where s1 ( t x) and s2 ( t y). t s 1 s 2 where t x s and t 1 y s 2 t s 1 s 2 where x t s and y 1 t s 2 t s 1 s 2 where t s1 ( x) and t s2 ( y) Therefore t t ( s1 s2) = ( t s1) ( t s2) ( x) ( y) Thus ( x y) ( x) ( y) and hence ( x y) ( x) ( y) International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 14 (3) (1) : Assume condition (3). Let F be a prime filter of R. We have to prove that R is dually normal. From Lemma 2.1.5, we have CF is a filter of R. Now, let x, y R and x y C( F). Then t x y is maximal for some t F. Now t x y is maximal xy t is maximal t ( x y) Since F is prime andt t1 t2 F, we get t 1 Suppose t 1 Similarly, ift 2 t ( x) ( y) [from (3)] t t 1 t 2 where t 1 ( x), t 2 ( y) t t t where 1 2 x t1, y t2 t t t where 1 2 t1x, t2 y F or t 2 F. F. Since t x is maximal and t F 1 1, we get x C ( F ). F, we get y C( F). Thus, we have CF is a prime filter of R. We conclude this chapter with the following theorem in which we characterize a dually normal ADL in terms of maximal elements. 2.3. Theorem : An ADL R is dually normal if and only if for x, y R, if x y is maximal then there exist x1, y1 R such that x x1, y y1 are maximal in R and x 1 y 1 0. Proof : Assume that R is dually normal. Let x, y R and x y is maximal in R.Then from Theorem 2.2, we have ( x ) ( y ) R. Now, 0 R 0 x1 y1 where x1 ( x ) and y1 ( y ) 0 x1 y1 and x x1, y y are maximal in R 1. That is, if for any x, y R such that x y is maximal, then there exist x1, y1 R such that x x1, y y1 are maximal in R and x1 y1 0. Conversely, assume the given condition. Let x, y R and x y is maximal in R. Now we prove that ( x ) ( y ) R. Since x y is maximal, from our assumption, there exist x1, y1 R such that x x1, y y1 are maximal in R and x1 y1 0. Clearly, x1 ( x ) and y1 ( y ) and hence 0 x1 y1 ( x ) ( y ) Now, 0 ( x ) ( y ) [0) ( x ) ( y ) R ( x ) ( y ). Therefore R ( x ) ( y ). Thus R is dually normal.
Dually Normal ADL REFERENCES [1] Rao, G.C. and Ravi Kumar, S., Minimal prime ideals in Almost Distributive Lattices, International Journal of Contemporary Mathematical Sciences, Vol. 4, 2009, no. 10, 475-484. [2] Rao, G.C. and Ravi Kumar, S., Normal Almost Distributive Lattices, Southeast Asian Bulletin of Mathematics. 32(2008), 831-841. [3] Swamy, U.M. and Rao,G.C., Almost Distributive Lattices, Journal of Australian Mathematical Society., (Series A),31 (1981),77-91. [4] William H. Cornish, Normal Lattices,J.Austral. Math. Soc. 16(1972), 200-215. [5] Ravi Kumar, S., Normal Almost Distributive Lattices, Doctoral Thesis (2010), Dept.of Mathematics, M.R.College(A), Vizianagaram. [6] Swamy, U.M. and Rao,G.C., Almost Distributive Lattices, Journal of Australian Mathematical Society., (Series A),31 (1981),77-91. [7] William H. Cornish, Normal Lattices, J.Austral. Math. Soc. 16(1972), 200-215. AUTHORS' BIOGRAPHY Lt. Dr. S. Ravi Kumar: He is presently working as Head & Associate Professor in the Department of Mathematics, Maha Rajah s Autonomous College, Vizianagaram. He obtained his Ph.D Degree on Normal Almost Distributive Lattices in Lattice Theory. His yeoman services as a lecturer, Head of the Mathematics Department, Controller of Examinations, Students Training & Placement Officer and as an Associate NCC Officer, made him a role model to thousands of students. He has one M.Phil in his credit. His papers more than 6 were published in various esteemed reputable Journals. He is a Member of Various Professional Bodies. G.Chakradhara Rao: He is working as a Professor in the Department of Mathematics, Andhra University (AU), Visakhapatnam which is a famous and reputed university of Andhra Pradesh. He obtained his Ph.D Degree in Mathematics from AU under the able guidance of Prof.U.M.Swamy. He has participated in many seminars and presented his papers. He has nearly 12 Ph.Ds and 13 M.phils to his credit. More than 66 articles were published in various National and International Journals. He authored 8 books on various topics in Mathematics. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 15